Trajectory Planning and Tracking Control for the Pendubot

In this paper, we study the trajectory planning and tracking control for the Pendubot, which is a two-link robot has a single actuator at the base joint of the first link. We realize a fast swing-down control under the sinusoidal-derivative (SD) controller which introduces a damper and spring, and obtain a desired trajectory of the state and the control input by using the swing-down controller and the time-reversal method. To track such a trajectory, we derive the linearized time-varying model of the Pendubot around the trajectory, and design a linear state feedback control along with a feedforward term determined by the trajectory of the control input. We study several obtained trajectories and focus on two issues: 1) the linear controllability of the corresponding linearized time-varying model for each trajectory; 2) the sufficient condition in the literature for checking the stability of the tracking control of the linearized time-varying model. Through our numerical investigation, we find that we have to choose appropriate control gains of the SD controller to guarantee the linear controllability, and that the sufficient condition is difficult to be satisfied.